Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Whole roots and powers

  1. Nov 28, 2011 #1
    [tex](a^{\frac{1}{2n}}a^{\frac{1}{2n}})^n=(a^{\frac{1}{2n}})^n(a^{\frac{1}{2n}})^n=(a^{\frac{1}{2}}) (a^{\frac{1}{2}})=a=(a^{\frac{1}{n}})^n[/tex]

    all above is just done by using that the order of the factors that you multiply does not matter

    we have proven that


    for any even number. Try saying that p is a odd number then n is a decimal number and taking the rooth with a number that is not whole does not make sense in a logixal approach of what one can really comprehend. For any even number of p what I did above makes sense. Could someone see an extention of this system. I want to make left side here:

    [tex]\sqrt[m]{\underbrace{a^{\frac{1}{n}} \, \cdot \, a^{\frac{1}{n}} \, \cdot \, . . . \, \cdot a^{\frac{1}{n}} \, }_{\text{m times}} \, }=a^{\frac{m}{n}} [/tex]

    to become right side by only using roths and powers of integers.
    Last edited: Nov 28, 2011
  2. jcsd
  3. Nov 28, 2011 #2


    User Avatar
    Science Advisor

    Certainly [itex]a^{1/2n}a^{1/2n}= a^{1/2n+ 1/2n}= a^{2/2n}= a^{1/n}[/itex].

    And [itex]\left(a^{1/n}\right)^m= a^{(1/n)m}= a^{m/n}[/itex]. Those are well known properties of the exponentials.
    Last edited by a moderator: Nov 28, 2011
  4. Nov 28, 2011 #3

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    If you know what Dedekind cuts are, the PlanetMath page http://planetmath.org/encyclopedia/ProofOfPropertiesOfTheExponential.html [Broken] does a nice job of extending exponentiation from the integers to the rationals, and then to the reals.
    Last edited by a moderator: May 5, 2017
  5. Nov 29, 2011 #4

    As far as I can see or from what I get from the proof at least


    I want to prove that


    What I have managed to show is that




    since n and p are integers to put p outside as own power goues without proving (could of course prove this
    and we get



    Last edited by a moderator: May 5, 2017
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook